THfi 


TfiUESGOPE-MlRROR-SCALE  METRO 


ADJUSTMENTS  AND  TESTS 


HOLMAN 


•  - 


THE 

TELESCOPE-MIRROR-SCALE  METHOD 

ADJUSTMENTS  AND  TESTS 


SILAS    W.    HOLMAN 

PROFESSOR    OF    PHYSICS    (EMERITUS) 
MASSACHUSETTS    INSTITUTE   OF    TECHNOLOGY 


Reprinted  from  The  Technology  Quarterly,  September,  1898 


NEW    YORK 

JOHN    WILEY   &    SONS 

LONDON  :    CHAPMAN  &  HALL,   LIMITED 

1898 


THE    TELESCOPE-MIRROR-SCALE   METHOD; 
ADJUSTMENTS  AND    TESTS.1 

BY  SILAS  W.  HOLMAN. 
Received  May,  1898. 

IN  the  telescope-mirror-scale  method  for  measuring  small  angles, 
an  illusory  estimate  of  the  accuracy  attained  may  easily  arise  through 
inattention  to  the  requisite  adjustments,  tests,  and  corrections.  Yet 
an  adequate  and  well-ordered  presentation  of  them  is  not  to  be  found. 
Partial  statements  are  given  in  text-books  on  manipulation,  but  Czer- 
mak2  alone  has  given  a  discussion  approaching  completeness.  Even 
this  omits  some  essential  points,  and  moreover  is  not  arranged  in 
a  manner  to  lend  itself  readily  to  the  practice  of  the  method. 

The  following  presentation  of  the  subject  is  designed  for  the 
observer  who  would  put  the  method  into  direct  service  for  obtaining 
measurements  of  a  specified  or  of  a  determinate  accuracy.  It  there- 
fore not  only  discusses  the  several  sources  of  error,  but  describes  the 
various  adjustments  and  tests  in  the  sequence  in  which  they  should 
ordinarily  be  made,  and  gives  a  numerical  measure  of  the  closeness 
with  which  each  must  be  carried  out  to  secure  a  specified  precision 
in  the  result.  Some  remarks  on  the  selection  of  instruments  are 
appended. 

The  general  plan  adopted  in  treating  each  adjustment  or  test  is 
as  follows  :  To  deduce  a  general,  though  usually  approximate,  expres- 
sion for  the  error  attending  its  omission,  or  preferably  for  the  correc- 
tion therefor.  To  deduce  therefrom  a  numerical  measure  of  the  close- 
ness with  which  the  adjustment  must  be  made  or  the  test  fulfilled,  in 
order  to  insure  a  designated  precision  in  the  use  of  the  method,  so  far 


1  Copyrighted,  1898,  by  S.  \V.  Holman.     Published  separately  by  John  Wiley  &  Sons, 
53  East  loth  Street,  New  York.     Cloth,  75  cents. 

2  CZERMAK.     "  Reduction  Tables  for  Readings  by  the  Gauss-Poggendorff  Mirror  Method." 
Besides  a  discussion  of  many  of  the  errors  of  the  method,  this  book  gives  extended  tables  of 
corrections  and  reductions,  of  much  service  in  long  series  of  observations.     The  text  is  in 
German,  French,  and  English,  in  three  parallel  columns. 


271629 


2  The   Telescope-Mirror-Scale  Method. 

as  that  source  of  error  is  concerned ;  and  to  make  certain  comments 
based  thereon. 

To  render  the  problem  definite,  the  tangent  of  the  angle  of  deflec- 
tion of  the  mirror  is  assumed  to  be  the  desired  result  of  an  observa- 
tion. The  deductions  are  easily  adaptable  to  other  less  frequently 
employed  functions.  The  expressions  for  the  errors  or  corrections 
are  brought  into  the  form  of  fractional  errors  or  corrections ;  that  is, 
they  are  expressed  as  a  fraction  of  the  value  of  the  tangent  of  the 
deflection.  The  correction  is  of  course  equal  to  the  error,  but  has  the 
opposite  algebraic  sign,  a  positive  error  requiring  a  negative  correction, 
and  vice  versa.  It  is  more  convenient  in  the  present  work  to  deal 
directly  with  corrections  than  with  errors.  The  numerical  solutions  are 
computed  on  the  basis  of  a  desired  precision  of  one  part  in  one  thou- 
sand, or  one-tenth  of  one  per  cent.,  in  the  resulting  value  of  the  tan- 
gent. The  precision  discussion  based  on  methods  elsewhere  stated 1 
is  sufficiently  obvious.  As  there  are  some  fifteen  sources  of  error,  the 
average  effect  to  be  assigned  to  each  consistently  with  the  prescribed 
limit  of  o.ooi  in  the  result  is  o.ooi  -f-  •*/  15  =  o.oo  030  nearly  enough, 
as  several  of  the  errors  are  usually  rendered  insignificant.  The  solu- 
tions are  also  made  for  a  scale  of  millimeters  at  a  distance  of  one  meter 
from  the  mirror,  and  for  a  maximum  deflection  of  500  mm.  It  will  be 
found  by  inspection  of  the  results,  especially  under  I,  III,  XII,  XIII, 
XIV,  XV,  that  o.  i  per  cent,  is  about  the  limit  of  accuracy  attainable 
under  these  conditions. 

In  the  employment  of  the  apparatus  absolute  values  of  the  tangent 
of  the  deflection  are  sometimes  sought,  in  which  case  the  measure- 
ments may  be  called  primary.  More  often,  however,  only  relative 
values  are  required,  or  rather  we  are  concerned  as  to  the  accuracy  of 
only  relative  values  of  the  tangents.  The  measurements  may  then  be 
called  secondary.  An  example  of  the  secondary  use  is  where  the  tele- 
scope and  scale  are  employed  with  a  reflecting  galvanometer  to  meas- 
ure currents,  and  where  the  "constant"  of  the  apparatus  is  found  by 
sending  a  known  current  and  reading  the  deflection.  In  such  cases 
any  constant  fractional  errors  in  the  telescopic  method  enter  into  the 
"constant,"  as  well  as  into  subsequent  observations,  but  with  opposite 
signs,  so  that  they  are  eliminated  from  the  results.  It  is  therefore 
needful  to  discuss  the  errors  with  respect  to  both  primary  and  second- 


1 "  Precision  of  Measurements."    John  Wiley  &  Sons,  New  York. 


Silas    W.  Holman.  3 

ary  use  of  the  method,  and  relief  is  thus  found  possible  in  the  latter 
from  some  of  the  exactions  of  the  former. 

When  reversals  are  taken,  that  is,  when  the  mirror  is  deflected  so 
that  a  reading  can  be  made  first  on  one  and  then  on  the  opposite  half 
of  the  scale,  certain  sources  of  error  are  reduced  by  averaging  the  two 
deflections.  This  is  usually  precluded  in  practice,  however,  by  such 
conditions  as  continual  change  in  the  reading,  avoidance  of  delay,  etc. 
Both  cases  must  therefore  be  discussed. 

The  desired  tangent  or  other  function  of  the  angle  of  deflection 
is  computed  from  the  observed  scale-reading  d  and  scale-distance  r  as 
stated  under  XVI.  We  will  assume  that  it  is  found,  with  due  allow- 
ance for  the  approximation  involved  (cf.  XVI),  from  the  expression 

tan  y  =  —  .  —  (i  -\-  u  -\-  v  -\-  w  -\-  .  .  .) 
2        r 

where  u,  v,  w,  etc.,  are  the  fractional  corrections  to  the  observed 
tangent  to  allow  for  the  various  sources  of  error  to  be  pointed  out. 
As  d  enters  as  a  direct  factor  in  this  expression  for  the  tangent,  the 
fractional  corrections  may  be  applied  directly  to  d.  In  fact  it  is  more 
convenient  in  deducing  the  formulae  to  find  an  expression  for  either 
Sd,  the  numerical  correction  to  d,  or  for  8d /  d,  the  fractional  correction 
to  d.  Thus  Sd  is  such  a  quantity,  expressed  in  scale  divisions,  that 
when  added  to  the  observed  scale-reading  d  it  will  give  the  correct 
scale  reading,  as  far  as  the  designated  source  of  error  is  concerned. 
And  &d I  d =  u  or  v,  etc.,  is  this  correction  expressed  as  a  fraction  of 
the  observed  reading  d.  The  algebraic  sign  of  any  correction  may  be 
either  determinate  or  indeterminate.  '  Also  it  may  be  noted  that  as  r 
enters  as  a  factor  in  the  denominator,  the  corrections  u,  v,  etc.,  may 
be  applied  to  or  deduced  for  r  instead  of  d,  if  desired,  but  with  the 
difference  that  the  algebraic  sign  would  be  reversed.  Any  corrections 
which  may  prove  to  have  a  constant  value  may  be  included  once  for 
all  in  the  numerical  constant  i  /  2r.  Constant  fractional  corrections 
disappear  when  merely  relative  deflections  are  used,  or  when  the  con- 
stant is  determined  by  calibration,  as  above  stated.  The  assumption 
is  made  that  none  of  the  errors  with  which  we  have  to  deal  are  in 
excess  of  i  or  2  per  cent.  Greater  ones  must  be  reduced  by  instru- 
mental rearrangement  before  they  can  be  determined  with  sufficient 
closeness. 


The   Telescope-Mirror-Scale  Method. 


The  algebraic  expressions  which  will  be  deduced  for  u,  v,  w,  etc., 
are  useful  in  two  ways  :  First,  they  enable  us  to  compute  the  correc- 
tion to  be  applied  to  any  observed  reading ;  second,  they  are  used  in 
the  precision  discussion,  which  shows  how  closely  each  correction  must 
be  worked  out  to  secure  a  prescribed  accuracy  in  the  value  of  tan,  cp  or 
how  close  an  adjustment  is  needed  in  order  that  the  correction  may  be 
omitted  without  introducing  more  than  its  due  share  of  error  into  the 
result.  These  will  be  called  negligible  corrections.  Any  error  of 
this  magnitude  produces  an  effect  not  more  than  is  admissible  on 
the  result,  and  the  effect  of  one  as  small  as  one-third  of  this  amount 
will  be  inappreciable. 

The  results  deduced  for  the  telescope-mirror-scale  method  are  in 
general  directly  applicable  to  the  mirror  method  where  a  beam  of  light 
replaces  the  telescope. 

THE  PROCEDURE. 

Preliminary. — 
Focus  the  cross- 
wires  in  the  eye- 
piece. Set  up  the 
apparatus  nearly  in 
proper  position  as 
in  Figure  i,  adjust- 
ing telescope,  mirror 
and  scale  so  that  the 
scale  is  visible.  The 
following  description 
will  assume  the  dis- 
position of  the  ap- 
paratus to  be  the 
most  usual  and  sim- 
ple one,  shown  in 
Figure  i  :  The  scale 
A,  B  is  horizontal 
with  its  middle  point 
just  below  or  above 

the  optic  axis  of  the  telescope,  which  is  perpendicular  to  the  scale. 
The  mirror  (plane)  faces  the  telescope  and  scale  at  a  convenient  dis- 
tance. The  horizontal  distance  r  between  scale  and  mirror  should  then 


-4 


P/ 


3» 


Side. 


FIG.  i. 


Silas    W.  Holman.  5 

be  roughly  measured,  say  to  I  per  cent.,  for  use  in  the  approximate 
computations  in  the  several  tests,  etc.  The  selection  of  this  distance 
is  determined  largely  by  convenience,  and  usually  lies  between  one 
and  three  meters.  It  must  be  sufficient  to  insure  to  the  smallest 
scale-readings  to  be  observed,  a  sufficient  number  of  divisions  to  be 
read  with  the  requisite  fractional  precision.  This  means  greater  dis- 
tances and  higher  magnifying  power  for  very  small  angles.  Labor 
in  computation  may  be  saved  by  making  this  distance  some  simple 
round  number,  e.g.,  1,000  or  2,000  scale  divisions,  but  the  work  of 
this  adjustment  will  usually  outweigh  that  of  computation.  The 
distance  must,  however,  be  maintained  accurately  constant.  The 
telescope  may  be  at  a  distance  different  from  that  of  the  scale, 
if  more  convenient,  but  further  removal  reduces  the  magnification. 
As  far  as  the  principle  of  the  method  is  concerned,  the  telescope 
may  be  at  any  position  whatever  relatively  to  the  scale,  consistent 
with  having  the  line  from  the  middle  of  the  scale  to  the  mirror  nearly 
horizontal,  as  pointed  out  in  test  X.  If  there  is  annoyance  from  dupli- 
cate reflections  from  the  glass  front  of  the  mirror  house,  the  glass 
may  be  slightly  tilted  forward. 

The  following  tests  and  adjustments  are  then  to  be  made  in  the 
order  in  which  they  are  given,  a  repetition  of  any  of  them  being  sub- 
sequently made  if  disarrangement  may  have  occurred  :  — 


ADJUSTMENTS  AND  TESTS. 

A  list  is  here  given  of  the  various  adjustments  and  tests,  together 
with  a  brief  statement  for  convenient  reference  of  the  closeness  neces- 
sary to  correspond  with  an  accuracy  of  one-tenth  of  one  per  cent,  in 
the  resulting  value  of  tan  cp,  assuming  a  straight  millimeter  scale  at 
a  distance  of  I  meter,  with  deflections  not  exceeding  500  mm.  and 
not  less  than  100  mm.,  readings  being  taken  to  tenths  of  a  millimeter 
by  the  eye. 

I.  CROSS- WIRE    Focus.  —  By  parallax   test    there    must    be   no 
apparent  motion  of  wires,  or  less  than  2L  division.     Telescope  must 
also  be  perfectly  focussed.     Primary  and  secondary  measurements  the 
same. 

II.  -  OPTIC    Axis    OF    TELESCOPE    RADIAL.  —  P.  M.     Cross-wires 
centered  within  i  mm.,  or  better,  to  0.2  mm.,  if  possible,  when  tested 


6  The   Telescope-Mirror-Scale  Method. 

on  scale  at  distance  of  i  m.     Optic  axis  must  pass  within  i  mm.  of 
axis  of  suspension. 

III.  SCALE  GRADUATION. — The  average  error  in  the  distance  of 
any  ruling  from  the  central  ruling  must  be  less  than  0.03  mm.     P.  M. 
and  S.  M.  the  same. 

IV.  COVER-GLASS   THICKNESS.  —  P.  M.    Glass  negligible  when 
less  than  i  mm.  in  thickness.     S.  M.     Glass  always  negligible.     Glass 
must  permit  good  definition. 

V.  COVER-GLASS    SURFACES    PARALLEL.  —  P.  M.     To  be  negli- 
gible, the  maximum  displacement  must  be  less  than  0.3  mm.  on  rotat- 
ing the  glass  180°  in  its  own  plane.     If  more  is  found,  the  glass  must 
be  kept  in  the  position  of  minimum  displacement.     S.  M.    The  same. 

VI.  COVER-GLASS   CURVATURE.  —  P.  M.    To  be  negligible,  the 
focal  length  of  the  cover-glass  must  exceed  70  meters.     This  will  be 
the  case  when  an  object  at  a  distance  of  10  meters  from  the  telescope, 
and  sharply  in  focus  with  the  cover-glass  interposed,  does  not  require 
to  be  moved  through  more  than  about  i  meter  towards  or  from  the 
telescope  to  maintain  the  focus  when  the  glass  is  withdrawn.     S.  M. 
The  error  is  here  always  negligible. 

VII.  MIRROR  THICKNESS.  —  P.  M.     Negligible  thickness  is  o.  5  mm. 
S.  M.   Always  negligible. 

VIII.  MIRROR  ECCENTRICITY.  —  P.  M.    Negligible  up  to  an  eccen- 
tricity of  10  mm.     S.  M.    Also  negligible. 

IX.  MIRROR  CURVATURE.  —  P.  M.    Correction  depends  on  amount 
of  eccentricity.     It  vanishes  with  no  eccentricity.     For  specific  case 
see  later.     S.  M.     Curvature  has  no  sensible  effect. 

X.  MIRROR  VERTICAL    AND  VERTICAL  ANGLE   TMO  SMALL.  — 
P.  M.    Telescope  and  scale  must  not  be  separated  vertically  by  more 
than  60  mm.     S.  M.     Negligible,  and  distance  need  not  be  very  small. 

XI.  SCALE  HORIZONTAL.  —  P.  M.     Error  insignificant  when  end 
of  scale  is  not  more  than  i  or  2  mm.  out  of  horizontal  through  middle. 
Negligible  up  to  12  mm.     S.  M.    Negligible  for  several  centimeters  of 
tipping. 

XII.  SCALE  PERPENDICULAR  TO  OM  AND  PLANE.  —  P.  M.    With- 
out reversals,  the  difference  in  distance  of  the  ends  of  the  scale  (the 
500  mm.  marks)  from  the  suspension  fiber  must  be  less  than  0.6  mm. 
With  reversals,  25  mm.  is  close  enough.     Scale  must  be  plane  within 
these  limits.     S.  M.    About  the  same  as  in  primary.     Hence  advan- 
tage in  reversals  where  practicable. 


Silas    W.  Holman.  7 

XIII.  NULL  POINT.  —  P.  M.    The  scale  once  fixed  in  place  can- 
not be  disturbed  without  impairing  XII.       Hence  accidental  change 
of  null  reading  must  be  remedied  by  turning  the  mirror,  not  by  mov- 
ing the  scale.     With  reversals,  the  null  reading  may  be  as  great  as 
17  mm.  without  correction.     Without  reversals,  the  null  reading  must 
be  less  than  0.6  mm.,  even  if  allowed  for.     S.  M.    Same  as  in  primary. 

XIV.  DISTANCE  OF  SCALE    FROM   MIRROR.  —  P.  M.    This  must 
be  measured  and  kept  constant  within  0.3  mm.     This  demands  more 
than  the  usual  attention,  and  renders  some  special  device  important. 
See  XIV  later.     S.  M.    The  same. 

XV.  ESTIMATION  OF  TENTHS    OF  DIVISION.  —  Nearest  tenth  is 
to  be  read  in  both  P.  M.  and  S.  M. 

XVI.  To  COMPUTE  THE  DESIRED  FUNCTION  OF  qp  from  the  ob- 
served deflection  d.     Details  are  given  under  XVI  later.-    Five  places 
of  significant  figures  should  be  used  in  d,  etc.1 

I.     CROSS-WIRE  Focus. 

The  cross-wire  intersection  must  be  brought  accurately  into  the 
focus  of  the  eye-piece  on  each  occasion  of  use  of  the  apparatus,  and 
by  each  observer  for  himself.  Inattention  to  this  point  may  easily  give 
rise  to  an  error  as  great  as  half  a  scale  division.  The  focussing  should 
be  done  by  the  parallax  method,  as  follows  :  Focus  the  wires  as  sharply 
as  possible  by  moving  the  eye-lens.  Then  focus  the  telescope  very 
carefully  on  the  scale  or  on  some  object  showing  some  sharply  marked 
point  of  reference.  Move  the  head  to  and  fro  sidewise,  so  that  the 
pupil  of  the  eye  shall  travel  from  one  side  to  the  other  of  the  aperture 
of  the  eye-piece.  If  the  wires  are  not  in  focus  they  will  appear  to 
move  over  the  scale.  If  so,  refocus  them,  and  then  refocus  the  tele- 
scope on  the  scale.  Continue  until  no  apparent  motion  of  the  wires 
is  perceptible.  Good  focussing  is  promoted  by  looking  away  from  the 
telescope  frequently ;  also  in  some  cases  by  fixing  the  attention  of  the 
other  eye  on  a  printed  page  held  at  the  distance  of  most  distinct  vision 
beside  the  telescope.  The  accurate  focussing  of  the  telescope  is  not 
less  important  than  that  of  the  wires.  The  error  from  imperfect  focus 
will  be  indeterminate  in  sign  and  magnitude. 

PRIMARY  MEASUREMENTS.  —  The  negligible  correction  will  then 
be  ±  8d,  whose  magnitude  must  be  such  that  8d / d  =  0.00030  for 


1  Computation  Rules  and  Logarithms.     The  Macmillan  Co.,  New  York. 


8  The  Telescope-Mirror-Scale  Method. 

the  smallest  value  of  d  to  be  used.  Now  deflections  of  less  than  100 
divisions  are  not  employed  in  exact  work  for  reasons  shown  in  XV. 
Hence  substituting  d  =  100  mm.  we  have  ±  §d  =  0.03  mm.  The 
extreme  motion  of  the  eye  in  the  parallax  test  will  obviously  produce 
a  displacement  of  the  maximum  error,  and  more  than  double  the  aver- 
age error.  Hence  the  focus  will  be  good  enough  when  the  maximum 
displacement  by  the  parallax  test  is  less  than  about  0.06  divisions,  or 
about  JQ-  °f  the  smallest  scale  division.  It  is  clear,  then,  that  the 
utmost  attention  to  this  detail  is  necessary. 

SECONDARY  MEASUREMENTS.  —  This  source  of  error  affects  pri- 
mary and  secondary  measurements  equally. 

II.     OPTIC  Axis  OF  TELESCOPE  RADIAL. 

This  adjustment  requires  two  operations  :  First,  centering  the 
cross-wire  intersection,  that  is,  bringing  it  into  the  optic  axis  ;  second, 
directing  the  optic  axis  towards  the  axis  of  suspension,  e.g.,  by  focuss- 
ing the  wire  intersection  upon  the  suspending  fiber. 

The  second  operation  is  obvious  enough,  but  must  be  repeated 
from  time  to  time  by  way  of  precaution.  It  is  not  easily  executed 
when  the  telescope  draw-tube  does  not  permit  of  focussing  on  objects 
as  near  as  the  mirror. 

The  well-known  method  of  centering  the  wires  is  briefly  as  follows  : 
Lay  the  telescope  in  a  pair  of  V  grooves  —  temporary  wooden  ones 
will  suffice.  Focus  on  the  scale,  or  on  a  few  rulings  roughly  equal  to 
the  scale  divisions,  placed  at  a  distance  about  equal  to  TM.  Rotate 
the  telescope  in  the  grooves.  The  wires  are  centered  when  the  inter- 
section shows  no  motion  on  the  scale  when  thus  rotated.  If  adjust- 
ing screws  are  provided,  the  diaphragm  carrying  the  wires  should  be 
centered  until  the  motion  is  less  than  o.  I  or  0.2  mm.,  as  demonstrated 
later.  If,  as  is  commonly  the  case,  there  is  no  means  of  adjustment 
of  the  wires,  they  must  be  remounted  if  the  extreme  apparent  dis- 
placement on  turning  through  180°  is  more  than  about  I  mm.,  for 
work  of  the  accuracy  here  assigned. 

DEMONSTRATION.  —  We  will  take  the  simplest  case,  where  the 
optic  axis  TF  of  the  telescope  is  at  right  angles  to  the  scale  AB.  If 
adjustments  XII  and  XIII  had  been  made  the  relative  positions  would 
be  slightly  different,  but  the  result  would  be  essentially  the  same  as 
far  as  this  source  of  error  is  concerned.  Suppose  the  telescope  so 


Silas  W.  Holman.  9 

located  that  TF  meets  the  mirror  at  c'  instead  of  c,  the  latter  being 
in  the  vertical  axis  of  suspension.  Let  cc'  be  denoted  by  s.  Then  we 
seek  an  expression  for  the  fractional  correction  Sd  J  d  for  the  error 
which  s  will  produce. 

If  c  were  at  c',  as  it  should  be,  and  were  deflected  through  an  angle 
y,  the  observed  reading  OA  would  be  the  true  reading  d  desired.  With 
c  at  c',  the  observed  reading  becomes  OA',  and  A'A  =  Sd.  As  Tc'  =  r, 
we  have  8d  :  c'  c"  =  d  :  r.  But  c1  c'1  =  z  tan  q>  =  2  (d  /  2r)  approx. 
Hence  §d  I  d  = 'zd  I  2r*  approx.,  for  a  deflection  on  the  side  consid- 
ered. For*-a  reverse  deflection  the  correction  would  obviously  be  of 
the  same  amount,  but  the  opposite  sign. 

Imperfect  centering  of  the  wires,  besides  misdirecting  the  optic 


FIG.  2. 


axis,  introduces  a  slight  error  through  the  smaller  magnification  of  the 
outer  scale  divisions,  but  this  may  be  easily  shown  to  be  small  com- 
pared with  the  foregoing. 

PRIMARY  MEASUREMENTS.  —  Without  reversals.  As  the  negligible 
correction  above  assigned  is  Sd  /d  =  o.oo  030,  the  negligible  value  of 
s  for  the  worst  case,  where  r=  1,000  mm.  and  d=  500  mm.,  is  to  be 
found  from  z  =  (Sd  /  d)  /  (d /  2r*}  =  o.oo  030  X  2  X  (i,ooo)2  -r-  500  = 
1.2  mm.  Thus  the  optic  axis  must  pass  within  about  ±  1.2  mm.  of 
the  axis  of  suspension,  which  demands  no  inconsiderable  care.  As 
the  adjustment  requires  two  operations,  both  must  be  made  closer 
than  this  limit,  or  one  must  be  rendered  imperceptible.  The  latter, 
by  careful  centering  of  the  wires,  is  usually  easier  of  accomplishment. 


IO  The   Telescope-Mirror-Scale  Method. 

If  this  is  done  so  that  the  displacement  in  the  above  test  is  less  than 
^  of  £,  i.e.,  0.4  mm.  on  a  scale  at  the  distance  TM,  or  better  to  0,1  or 
O.2  mm.,  this  part  of  the  error  will  be  negligible. 

With  reversal  of  deflections,  this  source  of  error  vanishes  when 
z  does  not  exceed  2  or  3  mm. 

SECONDARY  MEASUREMENTS.  —  Substantially  the  same  degree  of 
care  is  needed  as  in  primary  work. 

III.     SCALE  GRADUATION. 

The  graduation  must  be  uniform  to  such  an  extent  that  the  actual 
correction  §d  to  any  observed  deflection  d  shall  be  small  enough  to 
make  &d /  d  less  than  the  assigned  limit.  The  only  thorough  way  to 
test  this  is  to  measure  out  from  the  middle  point  of  the  scale  to  each 
successive  smallest  division  by  some  comparator  or  dividing  engine, 
thus  determining  every  value  of  §d /  d,  and  then  to  apply  the  correo 
tions  thus  found  unless  they  are  negligible.  If  an  error  not  exceeding 
J-0-  mm.  is  admissible  on  a  mm.  scale,  the  test  may  be  made  by  laying 
a  standard  (Brown  and  Sharpe)  mm.  scale  against  the  unknown  one, 
and  inspecting  the  coincidences  of  the  rulings.  The  first  50  or  100 
divisions  either  way  from  the  middle  are  not  employed  in  work  of  even 
moderate  accuracy,  because  the  value  of  d  is  then  so  small  that  the 
unavoidable  fractional  error  from  the  eye  estimation  and  other  sources 
is  excessive. 

PRIMARY  MEASUREMENTS.  —  The  requisite  accuracy  of  graduation 
for  the  above  assigned  limit  would  be  attained  when  for  the  smallest 
value  likely  to  be  used  for  d  in  such  work,  viz.,  </=  100  mm., 

*      7      

0.00030     .  •.  §d  =  0.030  mm.     That  is,  stated  briefly,  the  av- 
d   < 

erage  error  in  the  distance  of  any  scale  division  from  o  should  not 
exceed  0.030  of  a  division.  This  limit  can  hardly  be  attained  in  paper 
scales  without  great  care. 

SECONDARY  MEASUREMENTS.  —  Any  systematic  error  of  the  scale 
which  makes  it  too  long  or  too  short  by  a  constant  fractional  part,  i.e., 
for  which  &//d?is  constant,  will  be  without  effect,  since  it  will  cause 
the  value  of  the  " constant"  found  in  calibration  to  be  too  small  or 
too  large  by  an  equal  fraction,  so  that  the  error  in  the  constant  and  in 
deflection  readings  will  exactly  offset  each  other.  The  inequalities  of 
graduation  must  be,  however,  less  than  0.030  division. 


Silas  W.  Holman. 


II 


IV.     COVER  GLASS  THICKNESS. 

If  the  glass  has  plane  parallel  faces  and  is  of  a  thickness  t,  it  will 
displace  the  ray  parallel  to  itself  at  each  passage.  As  the  simplest 
case,  suppose  the  glass  to 
be  so  placed  that  a  ray 
OM  from  the  center  of 
the  scale  to  the  mirror  is 
normal  to  the  glass.  Then 
the  ray  will  experience  no 
displacement  in  passing 
from  O  to  M,  but  would  be 
displaced  from  MA  to  g  A' 
in  passing  from  M  to  the 
scale.  So  that  with  the  tel- 
escope at  O  a  reading  O  A' 
=  d  would  be  observed  in- 
stead of  the  true  reading  FlG<  3- 
O  A.  The  correction  &/i 

to  d\  is  therefore  A  A'  =gb.  Now  gb  =  be  —  ge,  be  =  t  tan  29,  ge 
=  t  tan  gee.     By  law  of  refraction,  n  being  the  index  of  refraction, 


6 


sin  h  gc 
*mffA'  ' 


i  ta.n^ce  .  „ 

=   ° —  approx.,  as  2(jp  is  small. 

n  tan  2(p 

=  t  (tan  2<y  —  —  tan  2qp)  approx. 
11 

=  t  (\ 1-\  tan  2<jp  approx. 

V  «  / 


For  ordinary  glass  n  =  — 

.-.  &/!=  L 
3 


d\ 


Inspection  of  the  diagram  shows  that  the  sign  of  the  correction  is 

positive  on  either  side  of  O.     Hence  . — .  =   —  .  —      The  reflecting 

d  3        * 

surface  of  the  mirror  is  supposed  to  contain  the  axis  of  suspension  in 

all  cases  unless  otherwise  specified,  so  that  O  M  =  r. 


12  The   Telescope-Mirror-Scale  Method. 

If,  however,  the  glass  is  not  perpendicular  to  O  M,  the  error 
becomes  unsymmetrical,  and  dependent  upon  the  horizontal  angle 
between  O  M  and  the  mirror.  It  is  best,  therefore,  that  the  angle 
should  be  kept  very  nearly  90°,  as  may  easily  be  done.  With  a  very 
thick  cover-glass  this  point  would  require  special  investigation. 

The  effect  of  the  thickness  of  the  cover-glass  may  be  readily  shown 
from  the  above  formula  to  be  equivalent  to  the  reduction  of  the  scale 
distance  r  by  one-third  of  the  thickness  of  the  cover-glass. 

PRIMARY  MEASUREMENTS.  —  The  requisite  thinness  of  the  cover- 
glass,  in  order  that  the  omission  of  the  correction  may  be  admissible, 
may  be  found  from 

—  =  . ~:   0.00030  .-.  t  =  3.1000   •  o.oo  030  =  0.9  mm. 

d  ?>r    < 

The  correction  must  therefore  be  applied  when,  as  is  almost  always 
the  case,  the  cover-glass  is  more  than  about  i  mm.  in  thickness. 

SECONDARY  MEASUREMENTS.  —  Since  the  fractional  correction 
§d I  d  is  constant,  the  correction  disappears  in  this  work  as  in  other 
similar  cases. 

V.     COVER-GLASS  SURFACES  PARALLEL. 

Defect  with  regard  to  this  is  common.  Its  effect  depends  so  much 
upon  the  various  possible  angular  positions  of  the  surfaces  relatively 
to  O  M  that  it  is  not  readily  reducible  to  a  general  expression,  but  an 
easy  and  sufficient  test  can  be  developed.  Give  the  mirror  a  deflection 
to  about  the  end  of  the  scale.  By  clamping,  or  in  some  effectual  way, 
hold  the  mirror  so  that  this  reading  shall  remain  fixed ;  focus  sharply ; 
then  using  due  care  not  to  disturb  the  apparatus,  place  the  cover-glass 
flat  against  the  objective  of  the  telescope,  or  better  against  some  dia- 
phragm arranged  to  hold  it  not  far  from  the  objective ;  read  closely. 
Now  rotate  the  cover-glass  through  180°  in  its  own  plane,  pressing  it 
against  the  objective  or  diaphragm,  and  read  again.  A  change  of  read- 
ing due  to  twice  the  refraction  by  the  prismatic  or  wedge  shape  of 
the  glass  will  be  found.  Turn  the  glass  in  its  own  plane  into  various 
(marked)  angular  positions  and  thus  locate  the  diameter  along  which 
the  displacement  is  greatest.  This  will  locate  the  direction  of  the 
edge  of  the  wedge,  that  is,  the  intersection  of  the  two  plane  surfaces, 
since  this  must  be  vertical  when  that  diameter  is  horizontal. 


Silas  W.  Holman.  13 

PRIMARY  MEASUREMENTS.  —  It  is  more  prudent  not  to  use  a  glass 
which  shows  a  maximum  displacement  6"  exceeding  0.3  mm.  at  a  great- 
est deflection  of  d  =  500  mm.  ;  for  the  negligible  correction  will  be 

Be?          i          S    — 

—  =  —  .    —  o.oo  030    .  • .  5  =  0.3  mm. 

d          2        500  < 

If  this  limit  is  not  easily  reached,  the  wedge  axis  may  be  placed 
horizontal,  which  will  sensibly  eliminate  the  error.  But  this  requires 
that  during  all  use  of  the  apparatus  the  cover-glass  be  continually 
inspected  to  see  that  it  is  in  the  proper  position. 

In  making  this  test  the  cover-glass  may  often,  to  good  advantage, 
be  rotated  in  position  instead  of  being  removed  and  placed  in  front  of 
the  objective.  In  that  case  special  care  must  be  taken  not  to  disturb 
the  mirror  when  the  glass  is  being  rotated  through  180°.  The  maxi- 
mum displacement  observed  in  this  method  of  test  in  either  case  is  in 
excess  of  the  worst  error  which  the  wedge  would  cause  in  the  obser- 
vations, unless  it  was  reversed  in  position  from  time  to  time  —  which 
must  be  guarded  against  if  the  glass  is  poor. 

SECONDARY  MEASUREMENTS.  —  No  relaxation  from  the  foregoing 
requirement  is  admissible. 

VI.     COVER-GLASS  CURVATURE. 

Minor  irregularities  in  the  surface  of  the  cover-glass  of  the  mirror 
house  produce  merely  a  blurring  of  the  image,  such  as  is  seen  in  look- 
ing through  ordinary  window  glass  with  a  telescope.  The  cover-glass 
must  be  sensibly  free  from  this.  If  either  surface  of  the  cover-glass  is 
systematically  curved,  the  glass  will  act  as  a  lens.  The  focus  of  points 
seen  obliquely  through  the  glass  is  then  changed  in  both  distance  and 
direction.  The  change  in  distance  is  either  unnoticed  or  is  corrected 
by  the  focussing  of  the  telescope.  The  change  of  direction  causes  an 
error  in  d.  The  glass  may  be  equivalent  to  a  spherical  lens  or  to  a 
cylindrical  lens.  The  effect  of  the  change  in  direction  may  be  studied 
by  the  central  ray  of  any  beam  ;  we  are  concerned  here  with  the  curva- 
ture as  revealed  by  a  horizontal  section  only.  For  simplicity,  suppose 
the  glass  to  be  equivalent  to  a  spherical  lens  D  G  of  focal  lengthy  with 
its  axis  coincident  with  O  M.  If  the  lens  is  convex,  the  true  read- 
ing A-  will  be  shifted  to  A'  toward  O ;  if  concave,,  in  the  opposite 
direction. 


14  The   Telescope-Mirror-Scale  Method. 

The  correction  required  and  the  method  of  test  may  be  developed 
as  follows  :  The  lettering  of  the  diagram  (plan)  being  as  before,  let 

O  A  =  d  =  true  deflection  reading, 

O  A'  =  d±  =  observed  deflec- 
tion reading, 

M  G  =  e  =p\  =  distance  of 
A'    cover-glass     G    from    mirror 
(roughly), 

J  and  M  =  conjugate  foci  of 

lens  G  when  MG  =  /,  the  focal 

length  of  G, 
FIG_  4-  JG  =p*  AM,A'J  =  straight 

lines. 

As  we  have  treated  separately  the  thickness  of  the  glass,  A  MO 
=  a,  A' JO  =  /3. 

tan  a  _  _  p<^  _      d       pi  -\-  r  —  e 
tan  ft         e  r  d^ 

d\  _       e     .      e 

d          r         /2 

But   by  the    law  of   lenses,  inserting  for  convenience   the   negative 
sign  because  /x  <  f> 

ill  ill 

—  —  —  =  —  or   —  —  —  =  — 

P\         Pi         f  *         P*         f 

Hence  multiplying  by  e  and  transposing  —  =    i   —  — , 


d± 
d 


The  signs  will  obviously  be  the  same  for  deflections  on  the  opposite 
side  of  O.     The  correction  is  therefore 


d  — 


d 


d\          e 


But    in    all  ordinary    cases  e  <  —  r,  so  that  for  the  present  purpose 

10 

—  is  negligible  compared  with   i,  and  — —  —  — . 

•/ 


Silas  W.  Holman.  15 

This  evidently  applies  to  deflections  in  either  direction,  provided 
that  the  axis  of  the  lens  coincides  with  O  M ;  otherwise  the  correction 
would  be  unsymmetrical  and  would  contain  another  term.  The  effect 
of  this  want  of  symmetry,  however,  would  be  detected  under  test  V 
for  wedge  shape  of  cover-glass,  and  therefore  need  not  be  here  discussed. 

If  the  cover-glass  proves  to  be  cylindrical  in  the  test  given  later, 
the  worst  effect  of  the  cylindrical  surface  will  be  produced  when  its 
axis  is  vertical.  The  correction  is  then  the  same  as  for  the  spherical 
surface. 

PRIMARY    MEASUREMENTS.  -  -  The    negligible    correction    may  be 

found  from   ~  —  —  0.00030. 

d    <    f 
Assuming  a  value  of  e  =  20  mm.,  not  often  exceeded,  this  yields 

f  =  20/0.00  030  =  7.  io4  mm.  —  70  m.  Hence  the  cover-glass,  whether 
equivalent  to  a  spherical  or  a  cylindrical  lens,  must  have  a  focal  length 
exceeding  70  m.  How  the  actual  value  of  f  is  best  determined  will 
presently  be  shown. 

SECONDARY  MEASUREMENTS.  -  -  The  fractional  correction  to  the 
deflection  is  constant.  Hence  in  secondary  work  the  error  introduced 
in  the  "constant"  by  neglect  of  the  correction  exactly  offsets  the 
errors  entering  into  the  subsequently  observed  deflections  by  the  same 
cause,  so  that  the  systematic  curvature  has  no  effect. 

MEASUREMENT  OF  f.  —  Direct  the  telescope  (any  good  telescope 
other  than  that  of  the  apparatus  will  answer  if  more  convenient)  upon 
any  well  defined  object,  such  as  a  printed  page,  held  at  a  distance  of 
several  meters  (io  or  more  if  practicable).  Interpose  the  cover-glass, 
placing  it  directly  in  front  of  the  telescope.  Focus  sharply ;  remove 
the  cover-glass.  If  the  glass  is  equivalent  to  a  lens  with  spherical 
surfaces,  the  object  will  cease  to  be  in  focus.  Without  changing  the 
focus  of  the  telescope,  bring  the  object  towards  it  (concave)  or  move 
it  away  (convex)  until  the  focus  is  again  sharp.  Let  a  denote  the  dis- 
tance from  glass  to  object  in  the  first  position  and  b  in  the  second. 

Then    — —  — ,   as  the  two  positions  are  conjugate  foci.     In 

oaf 

actual  measurements  it  is,  of  course,  better  to  measure  (a  —  b)  directly 
with  some  care,  and  then  a  roughly  (or  b),  thence  computing  b  and  f. 
To  detect  unequal  curvature  of  different  parts  of  the  glass,  it  is  well 
to  measure  f  for  each  of  its  four  quarters  successively,  covering  its 
remaining  surface. 


i6 


TJie   Telescope-Mirror-Scale  Method. 


It  may  sometimes  be  better  merely  to  test  whether  f  is  sufficiently 
great  without  actually  measuring  it.  Thus  if  the  limit  is  f  >  70  m., 
then  if  the  object  be  set  up  with  a  =  10  m.,  for  example, 


i    i      ,       i     80 

b          70          10          700 


•.  b  =  8.8  m. 


or    a  —  b  =  1.2  m. 

If,  therefore,  b  were  distant  from  a  by  as  much  as  i  m.,  the  glass 
would  be  good  enough.  Cover-glasses  should  never  be  found  so  poor 
as  this  on  good  instruments,  but  this  cannot  be  relied  upon,  as  makers 
frequently  send  out  very  unfit  glasses.  The  glass  should  be  tested 
by  focussing  through  it  separately  on  vertical,  horizontal,  and  oblique 
lines.  Difference  of  focus  shows  a  cylindrical  form  which  may  or  may 
not  be  superposed  upon  the  spherical.  The  test  should  be  applied  to 
that  value  of  f  which  makes  a  —  b  the  greatest. 


VII.     MIRROR  THICKNESS. 

The  mirror  usually  consists  of  a  thin  plane-parallel  piece  of  glass 
"  silvered  "  on  its  rear  surface.  If  in  any  case  the  mirror  were  very 
thick  it  would  require  special  investigation  as  to  corrections  for  want 
of  parallelism  or  planeness  of  faces ;  but  with  thin  mirrors  ordinarily 

used,  and  the  quality  which  is  insured 
xf  by  the  requirement  of  good  definition, 
x  ""  x  d  these  points  may  be  safely  neglected. 
Inspection  of  the  figure  will  show 
that  a  ray  a  b  passing  obliquely  into  the 
front  surface  of  the  mirror  will  pass  out 
along  c  d,  parallel  to  e  f,  the  path  it 
"^a  would  have  travelled  had  there  been  no 
glass  in  front  of  the  reflecting  surface. 
Also,  the  emergent  ray  c  d  is  displaced 
from  e ./by  the  same  amount  it  would  have  been  had  it  passed  through 
a  plane-parallel  glass  of  twice  the  thickness  of  the  mirror.  Since  by  the 
process  of  adjustment  the  line  O  Mis  sensibly  normal  to  the  mirror, 
the  correction  will  be  symmetrical  with  respect  to  O.  If,  then,  /'  rep- 
resents the  mirror  thickness,  the  correction  to  be  applied  to  it  will 

.    .       ,     ,          S  d          2          / 

obviously  be      =  —  .    — 

d  i         r 


FIG.  5. 


Silas  W.  Holman.  17 

PRIMARY  MEASUREMENTS.  —  Requisite  thinness  of  mirror  to  be 
negligible  =  0.5  mm. 

SECONDARY  MEASUREMENTS.  -  -  The  fractional  correction  being 
constant  may  be  wholly  neglected. 

VIII.     MIRROR   ECCENTRICITY.  , 

If  the  vertical  axis  of  suspension  does  not  lie  in  the  reflecting 
surface  of  the  mirror,  the  eccentricity  will  cause  an  error  in  the  scale 
reading.  As  the  simplest  case, 

let  c  M  represent  the  reflecting  ^Af 

surface    (plane)    in    its    normal  -^     -™- 

position,  O  M  being  perpendic-  \ 

ular  to   it ;   and  let    F  be  the  A/%  ° 

axis    of    rotation.       When    the 

mirror  turns  through  an  angle    Y 

qp  to  the  position  M',  the  re- 
flected ray  going  to  the  tele- 
scope at  O  will  be  A1  N  instead 
of  A  N  which  it  would  have  been  had  Y  passed  through  M.  The  cor- 
rection to  the  observed  reading  d'  =  O  A'  is  therefore  A  A' ;  and  for 
this  an  expression  must  be  found.  From  the  figure, 

'=  M  N  •  tan  2qr, 


Ar  0 

• 

FIG.  6. 


MN=eM  •  tan  qp  =  —  cM  •  tan  qc  =  JL    YM  •  tan2  <f, 

2  2 

tanqp=  —  tan  2  qp  (approx.  )    =  —  .  —  L.    Also  let    YM  =  X, 
2  2         r 

M  X       d* 

.  •  .    oa\    —     —   .    •  —  —  . 

8         r3 

This  correction  will  obviously  be  the  same  in  sign  and  amount  for 
equal  deflections  in  either  direction.     Hence 

SaT  i        X       dz 


PRIMARY  MEASUREMENTS.  —  The  requisite  smallness  of  X  may  be 
found  from 

Id  i          X        d*  — 

=  __  .    _       o.oo  030, 

d  8  r  r*    < 

• 

v  8  •  1000  •  iooo2 

•  *•  •«    =  -  —2  --  °-°°  °3°  =  10  mm. 


1 8  The   Telescope-Mirror-Scale  Method. 

It  is  rare  that  the  arrangement  of  the  apparatus  is  such  that  the 
eccentricity  exceeds  a  few  millimeters,  so  that  this  correction  is  ordi- 
narily quite  negligible.  If  the  normal  position  of  the  mirror  is  oblique 
instead  of  perpendicular  to  O  M,  the  correction  becomes  unsymmetri- 
cal.  and  would  require  further  investigation  for  special  cases  if  the 
eccentricity  were  large. 

If  the  apparatus  is  such  that  the  correction  is  not  small  and  must 
be  applied,  the  above  form  may  be  insufficient,  owing  to  the  inaccuracy 

produced  by  the  approximation  tan  qp  —  —  tan  2g>  employed  in  de- 

•I 

ducing  it.  It  is,  however,  easy  to  modify  the  expression  so  that  it 
may  have  any  desired  accuracy,  by  inserting  in  place  of  the  approxi- 
mation the  series  expressing  tan  qp  in  terms  of  d/r  given  later. 

SECONDARY  MEASUREMENTS.  —  As  the  fractional  correction  to  d 
has  a  constant  value  it  vanishes  for  all  values  of  d  as  in  former  cases. 

IX.     MIRROR  CURVATURE. 

This  produces  no  error  if  the  axis  of  suspension  of  the  mirror  is 
tangent  to  the  spherical  reflecting  surface.  If  the  curvature  of  the 
mirror  is  irregular,  the  definition  will  be  impaired  ;  but  if  the  telescope 
collects  the  light  from  the  whole  mirror,  no  sensible  irregularity  of 
reading  will  be  caused. 

In  the  diagram,  which  is  exaggerated  for  clearness,  let  Y  be  the 
vertical  axis  of  suspension,  MR  the  mirror  undeflected  (assumed  con- 
cave), and  O  Y  the 
line  from  F  to  the 
center  of  the  scale, 
and  to  which  the  un- 
deflected  mirror  will 
be  normal.  Suppose 
the  mirror  to  be 
turned  through  an 
angle  cp  about  Y, 
then  its  new  posi- 
tion will  be  M'  R', 
FIG.  7. 

M  having  moved  to 

M'  through  the  angle  M  YM'  =  y.  YM'  C  is  now  normal  to  the 
mirror,  and  let  M'  C  be  the  radius  p  of  curvature  of  the  mirror. 
A  ray  along  O  Y  would  now  meet  the  mirror  at  £,  whereas,  if  the 


Silas  W.  Holman.  19 

mirror  had  been  plane,  the  reflection  would  have  been  from  F  ;  and 
if  there  had  been  no  eccentricity,  from  M,  since  Y  would  then  have 
passed  through  M.  The  error  due  to  the  forward  motion  from  F  to 
E  would  be  capable  of  correction  by  the  same  expression  as  that 
for  M  F,  but  it  may  be  shown  to  be  so  small  as  to  be  negligible 
except  when  p  is  very  short,  in  which  case,  however,  the  mirror  is 
not  safe  for  use  in  accurate  work  even  with  the  application  of  the 
correction.  It  is  therefore  at  first  necessary  to  deal  only  with  the 
error  which  comes  from  the  increased  angle  between  the  normal  and 
O  E.  Owing  to  the  curvature  of  the  mirror,  the  normal  to  the  mirror 
at  E,  instead  of  lying  parallel  to  Ffand  therefore  making  an  angle  q> 
with  O  E,  makes  with  it  a  greater  angle  C  E  O  =  <f  -f-  a,  where  a  is  of 
course  equal  to  the  angle  between  the  tangents  at  J/and  M'  . 

We  desire  an  expression  for  the  fractional  correction  §d  f  d.     But 
&d        tan  2<Ti  —  tan  200        tan  2<Ti  —  tan  2qp 


— 
tan  2qpi  tan  2qp 

tan  <TI  —  tan  QD 

YJ.  ir 


approx. 


tan  q> 
Now  q>i  =  <p  -j-  a,  and  a  is  always  very  small  ; 

tan  OP  +  tan  a 

.  •  .  tan  cpl  =  ~       —  —  -  —  tan  <p  -f-  tan  a  approx. 
i  —  tan  qp  tan  a 

Sdi          tan  a 


d\  tan  qp 

But  as  M'  .  •.  £  and  £  YV~ are  parallel,  M'  CE=  a  ;  and  as  E  and  F 
are  nearly  coincident  and  F  M1  is  perpendicular  to  F  £7, 

tan  a         Jf 

tan  <p         /a  ' 

.  • .  -1   —  —  approx. 

*/i  p 

For  a  deflection  in  the  opposite  direction  the  correction  would  be 
the  same,  so  that  the  general  expression  for  the  correction  is 

£    J  V 

o  a          A. 
d  p 

For  a  concave  mirror  p  is  +>  for  a  convex  mirror  — . 

^PRIMARY  MEASUREMENTS.  —  The  limiting  value  of  p  may  be  found 
from 

8  ^i  /  «i  ^7  o.oo  030  =.  X I  p. 


2O  The   Telescope-Mirror-Scale  Method. 

Wjth  X  =  zero,  that  is,  no  eccentricity,  the  correction  vanishes.  For 
X  =  i  mm.,  p  =  3300  mm.  =3.3  m.  For  X  =  10  mm.,  p  =  33  m. 
and  so  on.  An  accidental  curvature  of  less  than  3  m.  is  not  unlikely, 
and  an  eccentricity  of  I  mm.  is  not  uncommon  and  by  no  means 
always  avoidable.  It  is  therefore  necessary  to  measure  p  roughly  as 
shown  below,  but  it  is  unlikely  that  the  correction  will  be  large  when 
a  presumably  plane  mirror  is  used,  and  a  little  consideration  of  the 
character  of  the  error  will  show  that  a  mirror  requiring  a  large  cor- 
rection cannot  be  employed. 

SECONDARY  MEASUREMENTS.  —  As  the  fractional  correction  is 
constant,  it  may  be  entirely  omitted. 

To  MEASURE  p.  —  Focus  the  telescope  sharply  on  the  reflection 
of  the  middle  of  the  scale  from  M  as  in  using  the  apparatus.  Meas- 
ure MO  within  a  few  mm.  Turn  the  telescope  slightly  so  as  to  be 
able  to  look  beyond  the  mirror,  and  place  a  printed  page  or  other  suit- 
able object  in  the  line  of  sight  at  a  distance  about  equal  to  MO. 
Without  changing  the  former  focus  of  the  telescope,  move  the  object 
towards  and  from  the  telescope  until  a  point  P  is  found  at  which  the 
focus  is  again  sharp.  Measure  M  P  within  a  few  mm.  Then  the 
radius  of  curvature  of  the  mirror  is 

MO 
P~  _MO 


MP 

in  which  the  numerical  values  of  both  MO  and  M  P  are  considered 
positive.  If  M  P  >  M  O,  clearly  p  is  positive  and  the  mirror  is  con- 
cave. If  MP  <  MO,  p  is  negative  and  the  mirror  convex. 

If  it  is  merely  necessary  to  know  whether  p  exceeds  a  specified 
limit,  this  expression  may  be  transformed  into 

MP  =  - 


2MO 

P 

It  is  then  merely  necessary  to  calculate  the  value  of  M  P  and  to 
see  fhat  the  object  is  in  focus  at  a  distance  less  than  this  in  the  above 
test. 

X.     MIRROR  VERTICAL  AND  VERTICAL  ANGLE   TMO  SMALL. 
Let  Mff=  horizontal  line  through  M, 

M  N=  normal  to  mirror, 
T  M  O  =  line  of  sight  when  q  =  o. 


Silas  W.  Holman.  21 

The  reflecting  surface  of  the  mirror  is  here  assumed  to  coincide 
with  the  axis  of  suspension,  so  that  M H '=  r. 

If  the  mirror  is  vertical  and  the  telescope  and  scale  are  both  at  H, 
no  correction  will  be  required.  The  mirror  may  be  made  vertical  or 
nearly  so,  but  T  must  usually  be  either  above  or  below  the  scale, 
hence  the  angle  T  M  O  cannot  be  zero.  Figure  8  represents  the  case 
when  neither  M  nor  T  M  O  are  zero,  and  where  N  O  <  N  T.  For 
this  case  the  fractional  correction  will  be  shown  to  be 

*d        NH  -    NO 


d  i* 

This  obviously  approaches  zero  as  N  If  approaches  zero,  /.  <?.,  as 
M  becomes  more  and  more  nearly  vertical.  It  will  disappear  if  the 
mirror  is  exactly  vertical  whatever  the  angle  T  M  O  (within  the  limits 
for  which  the  formula  holds).  But  as  it  is  of  course  impossible  to 
render  M  exactly  vertical,  it  is  necessary  to  inquire  how  nearly  so  it 
can  probably  be  rendered,  and  what  would  be  the  corresponding  limit 
of  T  M  O.  With  considerable  care  the  mirror  may  be  made  so  nearly 
vertical  that  A7"  will  fall  within  10  mm.  of  H. 

Inserting  then  NH=  10,   r=  1000,  and   Sdf  d~  0.00030  and 

solving  for  NO  gives  NO  =  o.oo  030  •   iooo2  -r-  10  =  30  mm. 

PRIMARY  MEASUREMENTS.  —  Thus  as  NO  is  approximately  equal 
to  N  Ty  the  telescope  and  scale  must  not  be  farther  apart  than  60  mm. 
Many  forms  of  telescope  and  scale  are  so  faulty  in  design  that  this 
closeness  of  approach  is  impossible.  Most  forms  provide  for  motion 
over  a  much  greater  range.  No  such  motion  is  necessary,  and  none  is 
desirable. 

SECONDARY  MEASUREMENTS.  —  Since  the  correction  is  a  constant 
one  so  long  as  T,  N,  and  O  remain  fixed,  no  correction  is  necessary  in 
secondary  work,  and  no  special  care  to  have  N  H  and  NT  small. 
Since,  however,  the  correction  formula  is  only  approximate  and  applies 
only  to  small  values  of  N  O,  the  adjustment  should  be  made  somewhat 
nearly  to  the  above  limit.  The  values  of  NH  and  NT  must  not 
change  during  a  series  of  measurements  by  as  much  as  the  above 
amounts. 

DEMONSTRATION.  —  Let  Figure  8  represent  the  apparatus  in  side 
view,  M . being  the  mirror,  O  the  middle  point  of  the  scale,  T  the 
telescope,  M  N  the  normal  to  the  mirror  when  deflected,  and  M  H 


22  The   Telescope-Mirror-Scale  Method. 

a  horizontal  line  through  M.  Let  the  dotted  line  through  O  be  a 
vertical  line,  and  assume  T,  N,  and  H  to  lie  upon  that  line,  so  that  all 
the  lines  of  the  figure  lie  in  a  vertical  plane  through  O  M.  The 
mirror,  which  should  be  vertical,  is  inclined  to  its  vertical  axis  of 
rotation  by  the  small  angle  a.  Thus  the  normal  M  N,  which  should 
coincide  with  J///when  undeflected,  makes  with  it  the  angle  N  M  H 
=  a.  The  telescope  and  scale  should  coincide  (T  with  O),  but  can- 
not. Hence  the  telescope  is  at  any  position  T  above  or  below  the 
scale  O,  and  N  M  T  —  N  M  O  =  /3. 

Let  Figure  9  represent  a  vertical  plane  through  the  scale  and 
viewed  along  M  H.  Then  5  5  will  represent  the  scale,  H'  H"  a 
horizontal  line  through  H  and  at  right  angles  to  M  H,  and  T',  N',  O', 
and  H'  the  points  T,  JV,  O,  and  H  respectively.  Suppose  now  the 


o      4 

O  '    'A'      V 


FIG.  8.  FIG.  9. 

mirror  is  turned  about  its  vertical  axis  of  suspension,  assumed  to  pass 
through  M,  through  a  small  horizontal  angle  g>.  Thus  the  normal 
M  N  will  describe  a  conical  surface  with  a  vertical  axis  and  having  its 
vertex  at  M.  This  cone  would  intersect  the  vertical  plane  through 
the  scale  in  an  hyperbola  with  its  vertex  at  M' .  Let  N'  N"  be  a  hori- 
zontal straight  line ;  then  if  N'  N"  be  the  horizontal  projection  on 
this  plane  of  the  part  of  this  hyperbola  described  when  M  turns 
through  qp,  we  have, 

tan  y  =  N'  N"  /  MH=  N'  N"  /  r. 

It  is  also  true  that  as  q>  is  small,  the  hyperbola  will  sensibly  coin^ 
cide  with  N'  N",  but  we  need  not  make  this  assumption.  Since  the 
reflected  ray  will  lie  in  a  plane  containing  T  and  M  N",  the  observed 
scale  reading'when  Mis  deflected  through  (p  will  be  at  A',  the  intersec- 
tion of  a  straight  line  T  N"  (prolonged)  with  the  scale.  Since  it  is 
read  by  a  horizontal  scale  having  vertical  rulings,  the  distance  O'  Af 


Silas  W.  Holman.  23 

along  the  scale  will  be  the  horizontal  projection  upon  the  scale  of  the 
parabolic  path  of  a  ray  from  T  reflected  by  M  upon  the  plane  of  the 
scale.  This  sh'ort  portion  of  the  parabola  of  the  ray  will  sensibly 
coincide  with  the  horizontal  straight  line  5  5. 

Let  A  represent  the  point  (unknown)  of  the  scale  at  which  the 
true  reading  corresponding  to  the  observed  reading  A'  would  fall; 
then 

&/!  =  a  A  —  a  A'. 

For  values  of  qp  so  small  that  tan  29  =  2  tan  q>  nearly  enough, 
&/!  =  2  r  tan  g>  —  O'  A'  =  2  N'  N"  —  O'  A'. 

But  also  for  larger  values  of  qp,  within  the  usual  limits,  although  the 
value  of  tan  2<p  exceeds  tan  <p  in  a  continually  increasing  ratio,  the 
value  of  O'  A'  increases  from  the  same  cause  in  sensibly  the  same 
ratio  ;  so  that  the  above  expression  for  8  d\  holds  nearly  enough  for 
all  cases.  From  the  diagram, 

O'A'  =  O'  K  +  KA'  =  N'  N"  -f  N"  K  •  tan  O'  T'  A'  =  N'  N"  + 

N'  N" 


Whence  as  2  N'  N"  =  d±  nearly  enough  (as  a  factor), 


dl  NT> 

But, 

NO         MO          r  cos  (a  -\-  ft)          cos   a   cos  ft  —  sin  a  sin    ft 

NT         M  T         r  cos  (a  —  ft)          cos   a  cos  ft  +  sin   a  sin    ft 

=  i  —  2  sin  a  sin  /8  approx.  when  a  and  /3  are  small ; 

NH  •  NO 
=  i  —  2    .       approx. 

8  </,         ^Y  •  7V~<9 
.  •.  _J_  = approx. 


The  correction  has  the  same  sign  on  either  side  of  O,  hence 
*d        NH  •    NO 


d 


approx. 


In  the  last  approximation  N  T  might  have  been  introduced  instead 
of  NO.     The  choice  is  determined  by  the  fact  that  O  and  ^Vlie  nearer 


24  The   Telescope-Mirror-Scale  Method, 

to  H  than  T  and  N,  so  that  the  approximation  is  closer  in  assuming 
sin  ft  =  N  O  I  r.  If,  therefore,  in  an  actual  case  the  telescope  is  much 
nearer  to  the  horizontal  than  the  scale,  it  will  be  slightly  better  to 
insert  N  T  instead  of  NO.  The  sign  of  the  correction  to  r  is  —  when 
N  O  <  N  T  and  +  when  N  O  >  N  T. 

XI.     SCALE  HORIZONTAL. 

To  adjust  the  scale  horizontal,  focus  T  on  the  scale,  with  .#f  swing- 
ing free.  Note  the  height  of  the  cross-hair  intersection  upon  the  divis- 
ions of  the  scale.  Deflect  M  \.o  the  right  and  left.  The  intersection 
should  remain  at  the  same  height.  If  it  does  not,  raise  one  end  of 
the  scale. 

If  the  scale  be  tipped  upward  or  downward  from  the  horizontal 
through  a  small  angle  7,  remaining  in  the  same  vertical  plane,  the 
reading  on  either  side  will  be  shortened  by  the  versed  sine  of  the 
angle  7 ;  that  is,  the  fractional  correction  to  d  will  be 

=  versm  7=1  —  cos  7. 


d 

PRIMARY  MEASUREMENTS.  —  For  the  requisite  closeness  of  adjust- 
ment, 

=  i  —  cos  7  ~~  o.oo  030, 


d  < 

.  • ,  cos  7  =  0.99  970,  and  7  =  i.°4. 

In  making  the  test  the  observed  change  in  the  height  Ji  of  the 
cross-hair  intersection  in  passing  from  the  middle  to  the  end  of  the 
scale,  would  be 

&/i  =  500  tan  7. 

Hence  the  requisite  closeness  will  be 

500  •   tan  7  =  500  •   tan  i.°4  =  12  mm. 

This  adjustment  can   be  made  with   perfect   ease  to   i   mm.  or  less, 

so  that  this  error  disappears. 

£  // 

SECONDARY  MEASUREMENTS.  —  As  is  constant  so  long  as  7  is 

d 

constant,  the  amount  of  tipping  of  the  scale  within  wide  limits  makes 
no  difference  so  long  as  it  remains  always  at  the  same  angle. 


Silas  W.  Holman. 


25 


XII.     SCALE  PERPENDICULAR  TO  O  M  AND  PLANE. 

To  render  M  O  A  a.  right  angle,  lay  off  or  select  two  points  A  and 
B  near  the  ends  of  the  scale,  exactly  equidistant  from  O.  Measure 
carefully  the  distances  M  A  and  M  B  from  the  suspension  fiber  of  M. 
Adjust  the  scale  until  the  two  are  equal.  When  this  and  the  preced- 
ing adjustment  have  been  completed,  the  scale  should  be  rigidly  and 
permanently  fixed  in  place,  and  means  provided  to  enable  the  fiber  to 
be  always  brought  back  to  its  present  position.  A  method  of  sup- 
port for  the  scale  quite  separate  from  the  support  of  the  telescope, 
as  described  later,  contributes  to  stability  and  convenience.  It  may 
be  noted  that  inequality  of  scale  readings  on  reversal  of.  M  is  not 
a  proof  of  imperfection  in  this  adjustment,  nor  equality  a  sufficient 
proof  of  correct  adjustment. 

The  fractional  correction  will  be  shown  to  be 

=  =F  sin  (3  tan  2  qp  —  —  sin2  /3  approx., 

d  2 

where  /3  is  the  small  angle  A  O  A'  by  which  the  scale  is  out  of  adjust- 
ment.     The   negative   sign   of   the   first   term   applies   to   deflections 
towards  A,  the   positive   to- 
wards B,  used  right-handedly. 

PRIMARY  MEASURE- 
MENTS.—  If  deflections  are 
taken  on  one  side  only,  i.  e., 
without  reversals  of  M,  the 
requisite  closeness  in  (3  will 
be  attained  when 

±  sin  /3  tan  2  <f  =  o.oo  030 


as  -I  sin2  /3  is  negligible  when 
the  apparatus  is  closely  ad- 
justed. For  r=  1000  mm., 
d  =  500  mm., 

rt 

sin  p  =  o.oo  030 


FlG.  I0. 


I  OOO 

500 


roughly  =  o.oo  060. 


This  corresponds  to  /3°  =  0.00060/0.0174 

=  o.°O35  roughly  =  2'. 


26  The   Telescope-Mirror-Scale  Method. 

But  it  is  more  convenient  to  have  the  requisite  closeness  expressed 
in  terms  of  M A1  —  MB'.  Now,  nearly  enough  when  /3  is  very  small, 
AA'  =  BB'=OA  •  tan/3=OA  •  sin /3  =  o.oo 060  •  500  =  0.30  mm., 
and  also  MA'—  MB'  =A  A' -\- A  £' =  2  •  OA  •  tan  /3  —  0.60  mm. 
Hence,  the  requisite  closeness  in  MA'  —  MB'  without  reversals  is 
0.6  mm. 

If  reversals  of  deflections  are  taken,  giving  d±  on  one  side  and  d^  on 
the  other,  the  corrected  deflection  is 

—  \d\-\-d-L  —  sin  /3  tan  2  qp  —  —  sin2  j8  -}-  sin  /3  tan  2(f  —  —  sin2  @\ 

2    L  2  2  J 

=  JL(4+4J)_  ±sin2/3. 

2  2 

Hence  the  requisite  closeness  will  be  attained  when 

—  sin2  /3'  =  o.oo  030,     .  • .  sin  /3'  =  0.025  . 

2 

This  corresponds  to  ft10  =  0.025/0.0174  =  i.°4  roughly.  Then 
2  '  OA  •  tan  /3  =  25  mm.  Hence  the  requisite  closeness  in  MA'  — 
MB'  witJi  reversals  is  only  25  mm.  This  demonstrates  a  great  ad- 
vantage in  reversals  in  primary  work  when  practicable.  For  not  only 
is  the  scale  more  readily  adjusted,  but  a  slight  undetected  deviation 
of  it  or  a  slight  lateral  displacement  of  the  suspension  is  much  less 
serious. 

The  scale  must  evidently  be  plane  within  the  same  limits  within 
which  it  must  be  perpendicular  to  O  M. 

SECONDARY  MEASUREMENTS.  —  In  these  nearly  the  same  closeness 
is  requisite  in  sin  /3  as  in  primary  work.  If,  however,  deflections  are 
restricted  to  the  outer  half  of  the  scale,  /.  e.>  250  to  500  mm.,  a  little 
less  closeness  will  suffice,  since  the  error  at  the  deflection  reading 
(whether  with  or  without  reversals)  at  which  the  calibration  is  made 
will  enter  into  the  value  of  the  constant,  and  this  will  eliminate  the 
error  at  the  same  point  in  subsequent  readings.  It  will  also  reduce 
the  average  error  about  one-half  if  the  calibration  deflection  is  about 
350  to  400  mm.,  but  the  difference  in  error  of  a  small  and  of  a  large 
deflection  will  still  remain  the  same. 

DEMONSTRATION.  —  Let  A  and  B  be  any  points  on  the  scale  equi- 
distant from  O  and  on  opposite  sides.  Then  O  A  and  O  B  will  be  the 
true  scale  readings  with  the  scale  properly  adjusted,  and  will  be  equal. 
O  A1  and  OB'  will  be  the  observed  readings. 


But 


Silas  W.  Holman.  27 

The  fractional  correction  8  d\  /  d\  on  the  side  towards  A1  will  be 

OA  —  OA'  _     OA 
OA'  ''  ~OAi 

O  A    sin  (90°  —  2  qp  —  p) cos  (2  y 


O  A!  sin  (90°  -f-  2qp)  cos  2  qp 


cos  2  qp  cos  /3  —  sin   2  or  sin  /3 

=  = 


•     /o  <. 
=  cos  £  —  sin  /3  tan  2  gc. 


cos  2  qp 
Now  cos  /3  =  (i  —  sin2 

=  i  —  —  sin2  /3  approx.,  as  '/3  is  very  small. 


1  1-2/0  •         /O     * 

L  =  -  _  —  i  =  —  —  sm2  /3  —  sin  p  tan  2  qp. 
' 


OA'  2 

Similarly  on  the  side  toward  B, 

O  R  T 

=  -  _  --    i   =  —  —  sin2  ft  +  sin  ft  tan  2  qp . 

C/  .Z5  2 


Or,  in  general, 


=  =p  sin  ft  tan  2  qp  —  —  sin2  /3. 
d  2 


XIII.     NULL  POINT. 

The  "null  point"  or  "zero  reading,"  i.e.,  the  reading  when  the 
mirror  is  undeflected,  must  be  the  middle  point  O  of  the  scale,  adjust- 
ment XII  having  been  made ;  and  0  M  must  lie  in  a  vertical  plane 
perpendicular  to  the  scale.  In  other  words,  the  line  of  sight  O  M, 
when  M  is  undeflected,  must  lie  in  a  vertical  plane  which  is  perpen- 
dicular to  the  scale  at  its  middle  point  O.  This  does  not  require  either 
that  any  normal  to  M  should  lie  in  or  parallel  to  that  plane,  or  that  the 
axis  of  the  telescope  should  lie  in  that  plane.  Or,  as  less  precisely 
expressed,  the  plane  of  the  mirror  need  not  be  parallel  to  the  scale, 
nor  the  axis  of  the  telescope  be  exactly  above  or  below  O.  The  tele- 
scope may  be  at  any  position  off  at  one  side  if  more  convenient,  but 
ease  and  accuracy  of  adjustment,  as  well  as  other  considerations,  lead 
to>  the  customary  location  of  the  axis  of  the  telescope  more  or  less 
exactly  above  or  below  O.  Provided  that  the  axis  of  rotation  (suspen- 
sion) and  the  reflecting  surface  of  M  are  sensibly  coincident,  and  that 


28  The   Telescope-Mirror-Scale  MetJwd. 

X,  XI,  and  XII  have  been  completed,  this  adjustment  may  be  accu- 
rately made  for  any  position  of  the  telescope  thus  :  Focus  the  tele- 
scope sharply  on  the  suspension  fiber  (axis  of  rotation)  of  M,  with  M 
swinging  free.  Turn  the  telescope,  or  shift  it  laterally  without  dis- 
turbance of  the  scale,  until  the  cross-hair  intersection  falls  upon  the 
fiber.  Still  without  disturbing  the  scale,  tip  the  telescope  about  a  hori- 
zontal axis  parallel  to  the  scale  until  the  cross-hairs  are  approximately 
central  on  the  mirror,  and  change  the  focus  until  the  scale  is  sharply 
defined.  The  reading  will  in  general  not  be  exactly  at  the  middle 
point,  but  the  telescope  and  scale  must  now  be  clamped  rigidly  in  posi- 
tion, and  all  subsequent  adjustment  of  the  null  reading  to  zero  must  be 
made  by  the  mirror,  that  is,  by  changing  the  direction  of  the  suspended 
system  until  the  null  reading  is  exactly  the  middle  point  O  from 
which  A'  and  B'  are  measured  off  in  making  adjustment  XII.  This 
will  be  effected  according  to  the  nature  of  the  apparatus;  e.g.,  by 
changing  the  directive  field,  in  a  sensitive  galvanometer ;  by  turning 
the  torsion  head  or  the  whole  instrument  if  it  is  an  electrodynamo- 
meter  or  electrometer,  or  by  twisting  the  mirror  upon  its  suspension 
rod,  and  so  on. 

If  the  axis  of  the  suspension  and  the  reflecting  surface  of  M  do 
not  coincide,  the  above  method  of  adjustment  becomes  inaccurate  to 
an  extent  depending  upon  the  eccentricity  of  the  reflecting  surface, 
and  on  the  departure  of  the  telescope  from  the  vertical  through  O. 
If  the  eccentricity  is  but  a  few  millimeters,  its  effect  is  wholly  negli- 
gible if  T  is  within  a  few  millimeters  of  this  vertical,  as  may  be  seen 
from  adjustment  VIII.  In  that  case  no  attention  to  the  adjustment 
of  T  and  O  beyond  casual  inspection  is  called  for. 

If  when  this  stage  of  the  adjustment  is  reached  it  is  found  that  the 
cross-hair  intersection  does  not  fall  at  the  right  height  upon  the  scale 
for  good  reading,  the  telescope  may  be  tipped  slightly  as  a  remedy. 
But  if  when  this  is  done  the  illumination  or  extent  of  field  is  not  all 
that  the  apertures  of  telescope  and  mirror  should  give,  then  the  adjust- 
ments must  all  be  repeated,  beginning  by  raising  and  lowering  the  tele- 
scope and  scale  together,  and  tipping  the  former  more  or  less,  focusing 
centrally  on  the  mirror  and  then  on  the  scale  alternately. 

During  observations  it  is  by  no  means  always  practicable  to  bring 
the  null  reading  exactly  to  the  middle  point  of  the  scale.  This  read- 
ing </0,  however,  is  taken  before  and  after  each  deflection,  or  with  suffi- 
cient frequency,  and  the  observed  deflection  reading  d'  is  corrected 


Silas  W.  Holman.  29 

for  dQ,  or  reversed  readings  are  taken,  giving  the  same  result.  The 
advantage  of  numbering  the  scale  from  one  end  continuously  to  the 
other  instead  of  both  ways  from  a  middle  point  is  apparent  in  this 
connection.  For  in  the  former  case  no  attention  to  the  sign  of  either 
d'  or  d0  will  be  necessary,  and  corrected  deflections,  viz.,  d!  —  dQ  will 
be  always  —  on  the  side  towards  the  zero  and  -f-  on  the  other  side,  the 
sign  taking  care  of  itself. 

But  even  with  the  correction  for  the  observed  null  reading  applied, 
there  remains  an  error  from  the  fact  that  the  ray  MdQ,  or  rather  the 
vertical  plane  through  it,  is  not  exactly  at  right  angles  to  the  scale. 
The  question  remains  how  closely  this  angle  must  approach  90°  ;  or 
in  other  words,  what  is  the  limit  of  displacement  of  the  null  reading 

when  corrected  for,  either 
j  by  observing  */0  or  by  re- 

versing. 

Suppose  the  scale  to 
'      be  properly  adjusted    at 
AB,  and    that    the    null 
reading   becomes    subse- 


^ 

~        ~*~  quently  displaced  (e.g.,  by 

change  of  torsion  in  the 
suspension,    of   direction 
of    field,     etc.)    from     O 
B  to  O". 

The  observed  reading 

towards  A  for  a  deflection  cp  of.  M  corrected  for  d^  (=  O  O")  will  be 
O"  C.  The  true  reading  would  have  been  O  A.  It  is  desired  then 
to  find  an  expression  for  the  fractional  correction  to  reduce  O"  C  to 
OA.  If  a  horizontal  scale  were  placed  perpendicular  to  MO"  at  a 
point  O'  such  that  MO'  =  MO,  then  the  observed  reading  O'  A'  on 
such  a  scale  would  be  equal  to  the  true  reading  O  A.  Imagine  also 
a  scale  O"  A"  parallel  to  O'  A'  but  drawn  through  O".  Then 

a  A'       o1  M       OM 

=  cos  p. 


O"  A"         O"  M  '  '    O"  M 

But  as  shown  under  XII  (case  of  OB /  OB'), 

• 

O"  A" 

— rt —  =  cos  /3  -f-  sin  /3  tan  2  cp 


30  The   Telescope-Mirror-Scale  Method. 

O'A' 


O"  C 

O  A 


=  cos2  /3  -f-  cos  ft  sin  ft  tan  2  <p 
=  i  —  sin2  ft  -\-  sin  ft  tan  2  r/>,  approx., 


O"  C 

as  with  ft  very  small,  the  factor  cos  ft  is  so  nearly  unity  as  not  sensibly 
to  affect  the  last  term  ; 

B  dl        OA  —  O"C 

.  •.  _*  •  =  —  sm2  ft  +  sin  ft  tan  2  (f 

and  in  general 

=  qp  sin  ft  tan  2  qp  —  sin2  ft, 

d 

the  -(-  and  —  signs  in  the  first  term  applying  respectively  to  right  and 
left-hand  deflections  (jp. 

PRIMARY  MEASUREMENTS.  —  With  Reversals  of  M.     Disregarding 

the  observational  sign  of  d,  the  mean  deflection  used  would  be  —  (d\  -f- 

2 

d%),  and  the  correction  would  be 

=  —  [sin  ft  tan  2  <JP  —  sin2  ft  —  sin  ft  tan  2  <$  —  sin2  /3] 


d  2 

or     =  —  sin2  ft  for  any  value  of  <f. 
The  requisite  closeness  of  ft  would  therefore  be  found  from 

8  d    =  .     o     r> 

.  o.oo  030  =  sm2  ft 

d    < 

.  •.  sin  ft  =  0.017  ;  ft  =  i.°o,  or 
O"  O  =  i  ooo  tan  ft  =  1 7  mm. 

Without  Reversals,  the  correction  would  depend  chiefly  on  <p  as 
ft  would  be  very  small  compared  with  any  practicable  value  of  qp. 
Hence  nearly  enough, 

ft  // 

=  =p  sin  ft  •  tan  2  qp« 

d 

The  requisite  closeness  of  ft  for  the  worst  case  where  r  =  1000  mm., 
d=  500  mm.,  would  then  be  found  from 

8d  —  500 

—  o.oo  030  =  ±  sin  ft   •  -? — 

d    <  1000 

.  •.  sin  ft  =•  o.oo  060  ft  =  o.°O35=    2',  or 
O  O"  =  i  ooo  tan  ft  =  0.6  mm. 


Silas  W.  Holman.  31 

Thus  with  reversals  a  rather  rough  adjustment  of  the  null  reading 
from  time  to  time  is  sufficient ;  but  with  deflections  on  one  side  only, 
even  with  the  null  reading  allowed  for,  this  reading  must  never  exceed 
about  half  a  millimeter  without  the  application  of  the  above  special 
correction  8  d\  /  d\  or  B  d%  /  d^.  It  is  also  essential  to  note  that  the 
adjustments  of  the  null  reading  must  be  made  by  turning  the  mirror, 
not  by  shifting  the  scale. 

SECONDARY  MEASUREMENTS.  —  The  requirements  here  are  pre- 
cisely the  same  as  in  primary  work,  and  the  frequent  and  often 
unavoidable  practice  of  reading  without  reversals  makes  the  above 
remarks  of  special  importance.  It  is  obviously  much  better  to  reverse 
where  practicable. 


XIV.     DISTANCE  OF  SCALE  FROM  MIRROR. 

This  is  invariably  the  shortest  horizontal  distance  from  the  axis  of 
suspension  to  the  vertical  plane  through  the  scale  ridings.  The  reflect- 
ing surface  of  the  mirror  should  lie  near  to  this  axis  when  possible, 
but  the  measurement  of  r  must  be  from  the  actual  suspension  horizon- 
tally to  a  vertical  line  through  the  middle  point  of  the  scale  after  XII 
has  been  performed.  It  is  important  to  note,  as  the  following  pages 
will  show,  that  most  of  the  adjustments  become  more  easy  and  the 
corrections  smaller  as  the  scale  distance  is  increased.  Therefore,  as 
scale  errors  are  easily  reduced,  it  is  better  to  make  r  as  large  (up  to 
3  or  4  m.)  as  is  consistent  with  the  magnifying  power  of  the  telescope, 
using  a  scale  of  not  more  than  i  m.  in  length.  The  telescope,  scale, 
and  instrument  containing  M  are  set  in  their  proper  positions  and 
completely  adjusted  before  r  is  measured.  This  is  then  carefully 
determined  by  a  horizontal  wooden  rod,  wire,  or  steel  tape,  using 
plumb  lines  if  necessary  at  M  or  at  5.  Before  being  used  in  compu- 
tations, r  must  be  expressed  in  the  same  unit  as  d.  The  telescope 
should  be  so  far  from  M  that  its  draw  tube  will  permit  focussing 
on  M. 

PRIMARY  MEASUREMENTS.  —  To  find  the  requisite  closeness  in  the 
measurement  of  r  we  have  to  note  that  r  enters  as  a  direct  factor  in 
the  denominator  of  tan  qp.  Hence  the  percentage  or  fractional  error 
in  tan  2  q>  is  proportional  to  that  in  r,  but  with  the  opposite  sign.  (See 


32  The   Telescope-Mirror-Scale  Method. 

"Computation  Rules,"  Proposition  I,  page  xii.)  The  limiting  value  of 
Sr / r  will  therefore  be  the  same,  neglecting  sign  as  8d /  d.  Therefore 

S  r  /  r  =  o.oo  030,  Sr  =  o.3mm. 

Hence  the  requisite  closeness  in  r  is  about  0.3  mm.  ;  that  is,  r  must  be 
measured  and  must  be  constant  to  about  0.3  mm.  See  remark  in  next 
paragraph. 

SECONDARY  MEASUREMENTS.  —  In  secondary  work  r  is  not  meas- 
ured, but  must  remain  constant,  and  suitable  means  must  be  provided 
to  see  that  it  is  so,  within  the  same  closeness  of  0.3  mm.  This 
requires  much  more  attention  than  is  usually  given  to  this  point, 
especially  when  the  mirror  hangs  on  a  long  suspension  so  that  change 
of  level  of  the  instrument  may  easily  change  r  by  I  or  2  mm. 

A  special  device  is  almost  a  necessity  in  careful  work,  to  facilitate 
the  test  for  constancy  in  the  distance  O  M.  In  many  instruments, 
especially  those  having  a  long  suspending  fiber,  a  slight  difference  of 
level  alone  produces  considerable  displacements  of  the  mirror  towards 
or  from  the  scale  or  laterally,  —  all  equally  objectionable.  The  direct 
application  of  XII  and  XIII  for  the  elimination  of  these  displacements 
on  each  day  of  use  of  the  apparatus  would  be  very  laborious.  This 
fact  together  with  inadvertence  frequently  leads  to  the  assumption 
that  the  distance,  once  adjusted,  remains  sufficiently  constant.  The 
foregoing  figures  as  to  the  requisite  closeness  show  the  danger  in 
such  neglect.  Devices  will  readily  suggest  themselves.  The  follow- 
ing will  sometimes  answer :  Place  the  points  of  the  leveling  screws 
in  positions  from  which  they  are  not  easily  displaceable.  Level  the 
instrument  until  the  suspension  swings  as  it  should.  Then  make  upon 
the  instrument,  near  the  lower  end  of  the  suspending  fiber,  say  on 
opposite  sides  of  the  suspension  tube,  reference  marks  so  located  that 
the  fiber  lies  in  the  line  of  sight  between  them.  Similarly  locate 
another  line  of  sight,  nearly  at  right  angles  to  the  first  and  also  pass- 
ing through  the  fiber.  For  convenience,  one  of  these  lines  should  be 
parallel  to  a  pair  of  the  leveling  screws,  and  also  to  the  scale.  At  each 
time  of  use,  adjust  the  screws  to  bring  the  fiber  into  both  of  these 
lines  of  sight.  Measure  O  M  once  for  all,  and  at  the  same  time  meas- 
ure between  two  points  chosen  for  convenience,  one  on  the  base  of 
the  instrument,  the  other  on  the  scale.  It  is  then  necessary  from  day 
to  day  merely  to  bring  the  fiber  into  the  reference  lines  by  turning  the 


Silas  W.  Holman.  33 

leveling  screws,  and  then  to  measure  the  distance  between  the  chosen 
points,  for  which  purpose  a  rod  cut  to  the  right  length  is  convenient. 

XV.     ESTIMATION  OF  TENTHS  IN  READING. 

Fractions  of  a  division  must  be  read,  and  this  can  be  done  only  by 
estimation  by  the  eye.  With  a  little  practice,  so  much  facility  in  the 
estimation  of  tenths  of  a  division  is  attainable  that  the  error  need 
never  exceed  one-twentieth  of  a  division,  with  an  average  error  of  half 
this  amount,  or  0.025  division. 

PRIMARY  MEASUREMENTS.  —  The  limit  of  attainable  accuracy 
being  fixed  at  ±  d=  0.025  division,  and  the  desired  fractional  accu- 
racy being  ±  &  d  /  d  =  0.00030,  the  minimum  admissible  value  of  d 
will  be  d=  0.025  /  0.00030  =  83,  or  in  round  numbers  d=  100  divi- 
sions, whatever  the  size  of  the  division.  Smaller  deflections  than  100 
must  not  be  used  in  the  most  careful  work,  and  preferably  not  less 
than,  say,  200  divisions,  whatever  the  scale  distance. 

SECONDARY  MEASUREMENTS.  —  Same  as  in  primary. 

XVI.     To  FIND  qp,  TAN  qp,  SIN  qp,  ETC.,  FROM  d  AND  r. 

In  the  application  of  the  telescope  and  scale  method  it  is  desired 
to  find  from  the  observed  values  of  d  and  r  the  corresponding  values 
of  qp,  tan  qp,  sin  qp,  etc.,  according  to  circumstances. 

For  the  most  accurate  work,  especially  in  primary  measurements, 
the  best  way  is  to  compute  tan  2  qp  =  (d  /  r)  or  log  tan  2  qp,  thence  by 
tables  of  natural  or  log  tangents  to  find  2  qp.  This,  of  course,  gives  qp  at 
once,  and  tan  qp,  sin  qp,  and  other  functions  can  then  be  found  from 
tables.  In  some  cases,  notably  in  secondary  work,  it  is,  however, 
more  convenient  to  have  an  expression  for  qp,  tan  qp,  or  other  function, 
directly  in  terms  of  d  and  r.  Such  expressions  generally  take  the 
form  of  a  series.  Several  may  be  found  in  Czermak.  The  expression 
for  tan  qp  is  the  one  in  frequent  use,  and  therefore  this  alone  will  be 
given  here.  It  is  derived  from  a  development  of  tan  qp  in  series  with 
ascending  powers  of  tan  2  qp,  viz.  : 

tan  qp  =  —  tan  2  qp  |    i  —  —  (tan  2  qp)2  -f-  _  (tan  2  qp)4  —  .  .  .  .   J. 

Substituting  d  /  r  for  tan  2  qp  gives 

1  d  r  i    ,  d  \2    -      i    /  d  x4  -| 
tan  qp  =  — -    •    —      I  —  —  (  —  )    4-  —  ( — )  —  ....    I. 

2  r  |_  4e\rJ  %   \  r  ) 


34  TJie   Telescope-Mirror-Scale  Method. 

This  could  be  used  directly,  but  may  be  modified  into  a  more  conven- 
ient form.  Multiplying  the  terms  of  the  parenthesis  by  d  gives 

i     r  ,        i        d*          i         d^  -| 

tan  y  = \d  —  —  •    — —  -\ •  — ..... 

2r  L  4        r*  r*  J 

Then  the  quantity 

_L      fll-L  JL        ^L 
'  ~4        r*   '   '  T          r* 

being  very  small  may  be  conveniently  treated  as  a  correction  £  to  be 
applied  to  d.  So  we  may  write 

tan  op  =  \d  -r-  8] . 

2  r 

The  procedure  would  be  to  compute  beforehand  a  table  of  values 
of  this  correction  for  the  given  value  of  r  and  for  successive  values  of 
d  (=  200,  250,  300,  .  .  .  500)  at  sufficiently  short  intervals  over  the 
desired  range.  Then  for  a  subsequent  observed  value  of  d',  the  cor- 
responding value  of  &'  would  be  taken  by  interpolation  in  a  table  (or 
upon  a  plot),  and  applied  to  d'.  This  corrected  value  of  d'  would 
then  be  multiplied  by  i  /  2  r  to  obtain  tan  (p.  As  the  correction  is 
small,  only  an  approximate  value  of  r  is  required  in  computing  it,  and 
slight  changes  in  r  during  the  progress  of  work  would  not  vitiate  the 
table,  although  they  must  be  allowed  for  in  the  term  i  /  2  r. 

In  secondary  work  any  of  the  above  methods  of  finding  gr,  tan  <p,  etc., 
may  be  used,  but  for  use  in  connection  with  tangent  instruments  the 
application  of  the  series  expression  is  especially  advantageous.  This 
will  be  illustrated  by  the  tangent  galvanometer.  For  this  instrument 
the  current  C  is  related  to  the  steady,  angular  deflection  o;  which  it 
produces,  by  the  expression 

C  =  k  tan  <p, 

where  k  is  a  numerical  factor  which  is  constant  so  long  as  the  magnetic 
conditions  and  dimensions  of  the  instrument  are  constant.  Substitut- 
ing the  above  expression  for  tan  qp  gives 

C  =  —  [d  +  B] 
2  r 

or,  as  r  is  a  constant  throughout  the  work,  we  may  write 


Silas  W.  Hoi-man.  35 

where  K  is  a  numerical  constant  =  k  /  2  r    and 

8-         _L      ^l-4_  _L       — 
'  ~4    '   ~r*  ~  '  T        r4 

But  in  the  employment  of  such  a  secondary  instrument  it  is  calibrated, 
that  is,  its  constant  is  found,  by  sending  through  it  some  known  cur- 
rent C'  and  observing  the  steady  deflection  d'.  Then  denoting  by 
8'  the  known  value  of  the  correction  8  computed  for  this  deflection  d', 

we  have 

C' 
K  = 

d'  +  8'  ' 

This  gives  us  the  numerical  value  of  K  so  that  C  can  be  at  once  com- 
puted for  any  subsequently  observed  values  of  d  by  the  expression 

C=K[d+Z\, 

the  proper  value  of  8  being  taken  from  a  table  or  plot  as  above  de- 
scribed. The  correction  8  will  be  a  quantity  to  be.  subtracted  from  d, 
and  therefore  expressed  in  the  same  unit,  e.g.,  millimeters.  The  table 
must  be  carried  out  to  the  next  place  of  significant  figures  beyond  the 
last  place  obtained  in  reading  the  scale,  e.g.,  to  hundredths  of  mm.  if 
readings  are  taken  to  tenths  of  mm.  It  must  also  be  computed  for 
sufficiently  short  intervals  to  enable  interpolation  to  be  made  with  cor- 
responding closeness,  i.  e.,  to  o.oi  mm.  or  0.02  mm.  in  the  above  case. 
And  if  a  plot  is  used  it  must  be  on  a  sufficient  scale  and  with  suffi- 
ciently frequent  points  to  enable  this  same  closeness  to  be  obtained. 
Unless  many  readings  of  d  are  to  be  reduced,  it  is  as  well  to  compute 
8  for  each  observation  as  to  make  a  table.  Extensive  tables  of  8  for 
graded  values  of  r  and  d  are  given  in  Czermak.  The  number  of  terms 
to  be  retained  in  the  series  in  computing  8  must  be  determined  by 
computing  for  the  largest  value  of  d  the  value  of  the  successive  terms 
until  one  is  reached  which  is  negligible,  e.g.,  less  than  about  o.oi  mm. 
in  the  above  case. 

If  the  instrument  is  used  always  with  deflections  near  the  value  d', 
observed  in  calibrating,  it  is  evident  that  the  values  of  8  both  in 
obtaining  K  and  subsequent  measurements  will  be  nearly  the  same. 
Hence  if  8  were  omitted  in  computing  K,  and  also  in  all  subsequent 
computations  of  C,  the  errors  thus  introduced  would  nearly  offset  one 
another.  To  see  how  wide  a  range  of  deflections  could  thus  be  used 


3^  The  Telescope-Mirror-Scale  Method. 

without  introducing  into  C  a  resultant  error  exceeding  the  limit  which 
we  have  been  using  in  the  preceding  discussion,  we  have  merely  to 
find  the  two  deflections  d±  and  dz  at  any  desired  part  of  the  scale  for 
which  (S2  —  Sj)  /  d±  =  o.oo  030.  For  a  rather  extreme  case  of  d-^  = 
400  mm.,  r=  1000  mm., 

S2  —  Sj  =  o.  12  mm. 

The  value  of  dz  for  which  S2  would  be  S400  -f-  o.  1 2  could  be  found 
from  tables,  but  in  default  of  these  we  may  compute  as  follows,  neg- 
lecting the  second  term  in  8, 

i        43          i        d* 

-  '  ~\ •—¥••=  °-12 

4         H  4         r* 

.-.  d*  —  d*  =0.48  -    io6 
.  •.  d£  =  4003  +  0.48  •   io6 

=  64  •  48  •   io6 
.*.    d<i  =401.0  mm. 

Thus  the  correction  could  be  neglected  in  this  case  only  over  a  range 
of  i  mm.  Even  with  large  values  of  r  and  smaller  ones  of  d  the 
correction  becomes  negligible  over  only  a  centimeter  more  or  less,  and 
thus  is  practically  never  negligible  in  o.  i  per  cent.  work. 

SELECTION   OF   APPARATUS. 

Rigidity  of  construction  is  a  prime  requisite.  It  is  often  impaired 
by  unnecessary  and  weak  adjustments  or  poor  clamping  devices.  The 
scale  may  be,  and  generally  is,  carried  on  the  same  support  as  the  tele- 
scope. But  an  entirely  independent  support  is  to  be  preferred,  as 
facilitating  adjustment  and  promoting  stability.  This  will  be  the  more 
obvious  if  it  be  remembered  that  the  telescope  need  not  be  exactly  over 
the  center  of  the  scale  or  in  any  determinate  position  relatively  to  it, 
although  an  approximately  central  position  is  usually  more  convenient 
and  is  presupposed  in  the  preceding  discussion  of  some  of  the  cor- 
rections. A  convenient  method  of  supporting  the  scale  would  be  by 
a  bracket  at  each  end  clamped  to  the  table  by  screws  passing  through 
slots  in  the  bracket.  This  would  permit  the  needed  forward  and  back 
motion  of  the  ends  separately.  Long  and  wide  slots  in  the  vertical 
arms  of  the  brackets  would  afford  the  necessary  vertical  and  endwise 


Silas  W.  Holman.  37 

range  of  adjustment.  With  this  arrangement  the  scale  can  be  placed 
in  final  adjustment  and  clamped  in  position  without  employing  the 
telescope,  and  the  latter  may  be  separately  put  in  place  with  the  mini- 
mum of  trouble,  and  with  no  chance  of  disturbance  of  the  scale. 

The  telescope  must  have  rotation  about  both  vertical  and  horizon- 
tal axes.  It  should  be  provided  with  a  vertical  adjustment  through 
a  range  of  six  inches  or  more,  preferably  by  a  round  rod,  which  gives 
at  the  same  time  the  needed  vertical  axis.  The  base  must  be  provided 
with  screw  holes  for  immovable  attachment  to  the  table.  The  draw 
tube  of  the  telescope  must  be  thoroughly  firm,  so  that  a  moderate 
lateral  pressure  on  the  eye-piece  shall  produce  no  permanent  shifting 
of  the  cross-wires  over  the  scale.  The  tube  must  be  long  enough  to 
focus  on  objects  at  a  distance  of  somewhat  less  than  one  meter.  The 
definition  of  the  telescope  should  be  tested  as  carefully  as  desired,  but 
will  usually  be  sufficiently  good  if  a  clear  image  of  the  scale  is  given 
under  fair  illumination  at  the  usual  distance. 

The  definition  of  the  mirror  on  any  instrument  to  be  used  with  this 
method  must  be  tested,  for  example,  by  the  quality  of  the  image  of 
the  scale  when  viewed  through  the  telescope  at  the  usual  distance,  but 
with  no  interposed  cover-glass.  The  effect  of  the  cover-glass  on  the 
definition  must  be  carefully  tested  by  alternately  inserting  and  with- 
drawing it,  noting  the  change  in  sharpness  of  the  image  of  the  scale. 
The  best  of  thin  plate  glass  is  barely  good  enough,  and  selection  from 
samples  is  often  necessary,  especially  to  secure  sufficiently  parallel 
surfaces.  There  is  little  advantage  in  having  the  diameter  of  the 
objective  of  the  telescope  more  than  double  that  of  the  mirror  to  be 
used  ;  an  inch  to  an  inch  and  a  half  is  usually  abundant,  and  three- 
quarters  of  an  inch  is  often  enough.  A  magnifying  power  of  twelve 
to  fifteen  diameters  is  desirable  but  is  not  common.  The  relation 
between  size  of  scale-division,  magnifying  power,  and  distance  will  be 
briefly  considered. 

Let  O  M  =  r,  u  =  magnifying  power,  and  SQ  =•  the  best  size  of 
division  upon  which  to  estimate  tenths  of  a  division  by  the  unaided 
eye  ;  this  is  about  I  mm.  Then  SQ  /  v  is,  nearly  enough,  the  angle  in 
radians  subtended  by  one  scale  division  when  seen  directly  at  the 
distance  v  of  most  distinct  vision.  If  at  any  other  distance,  as  at 
TM-\-MO,  reckoning  T  M  from  the  eye-piece,  the  angle  becomes 
y0  (T M -\-  MO}.  This  is  the  distance  in  the  telescopic  method.  In 
order  then  that  the  telescope  shall  compensate  for  this  removal,  that 


38  The  Tclcscopc-Mirror-Scale  MetJiod. 

is,  in  order  that  it  shall  make  the  arrangement  as  sensitive  as  a  direct 
reading  at  the  distance  of  most  distinct  vision  (which  is  the  condition 
attainable  by  the  use  of  a  spot  of  light  proceeding  from  T  and  reflected 
from  M  to  the  scale  at  A),  the  telescope  must  have  a  sufficient  magni- 
fying power.  This  must  be,  for  objects  at  a  distance  TM-\-MO, 
such  that  SQ  /  (T M  -f  M  O)  =  SQ  /  v,  or  u  =  (TM  +  MO}  /  v.  For 
the  case  where  TM  =  MO  =  r=  1000.  mm.,  and  v  =  250.  mm., 
this  yields  u  =  8.  Telescopes  are  often  furnished  for  this  use  with 
smaller  values  of  n  than  that  just  deduced,  hence  for  ordinary  use 
with  low  power  instruments,  the  millimeter  is  about  the  proper  size 
of  scale  division,  but  this  is  by  no  means  the  case  with  higher  pow- 
ers. The  best  value  of  s  for  any  fixed  value  of  u  would  be  such  that 
«  s  I  (T  M  -f  M  O}  =  SB  I  v,  or  s  /  SQ  =  (TM  +  MO)  /  u  v.  For  a 
very  good  glass  u=  15,  so  that  with  the  values  as  before,  the  scale 
should  be  in  half  millimeters  if  used  at  the  distance  of  r  =  I  meter ; 
since  s  /  SQ  =  2000  /  ( 1 5  X  2  5)  =  o.  53. 

With  the  mirror  and  spot  of  light  method,  the  percentage  or  frac- 
tional precision  with  which  a  given  angle  can  be  measured  with  the 
scale  at  a  distance  M  O  is  proportional  to  that  distance ;  that  is,  the 
precision  for  M  O  is  to  that  for  a  distance  unity  as  MO  :  I.  With 
a  telescope  of  magnifying  power  u  the  gain  is  further  increased  by  the 
telescope  in  the  ratio  S  /  s,  where  5  is  the  length  of  division  actually 
employed,  and  s  is  the  best  length  computed  as  just  indicated,  with 
the  limitation,  of  course,  that  5  is  greater  than  s.  Thus  by  the  em- 
ployment of  unduly  long  divisions,  the  advantage  derivable  from  high 
magnifying  power  may  be  sacrificed  and  most  of  the  advantage  over 
a  much  cheaper  instrument  rendered  idle.  Of  course  the  superiority 
of  the  telescopic  method  over  the  spot  of  light  does  not  lie  wholly  in 
the  magnification,  but  partly  in  better  definition,  and  in  the  avoidance 
of  the  necessity  for  screening.  On  the  other  hand,  the  latter  method 
has  the  merit  of  simplicity  and  cheapness,  as  well  as  of  facility  of  read- 
ing where  there  is  much  jarring  and  high  accuracy  is  not  demanded. 

As  to  materials  for  the  scale,  a  white  metal  surface  with  fine  black 
rulings  would  be  best,  and  is  almost  indispensable  in  accurate  work, 
but  is  expensive  and  not  generally,  if  at  all,  offered  by  makers.  White 
porcelain  or  glass  with  fine  black  rulings  is  the  next  choice.  Paper  on 
wood,  or  celluloid,  is  not  to  be  relied  upon  in  careful  work,  and  must 
be  thoroughly  tested  for  uniformity. 

The  warping  of  wooden  scales  may  introduce  serious  error  (cf.  III). 


Silas  W.  Holman.  39 

The  numbering  usually  extends  from  zero  at  the  middle  towards  each 
end,  but  for  many  purposes  a  continuous  numbering  from  one  end  is 
more  convenient  (cf.  XIII).  By  the  use  of  a  circular  scale  with  its 
center  of  curvature  at  M,  the  readings  become  directly  proportional  to 
twice  the  angle  of  deflection,  and  the  focus  is  equally  good  throughout 
the  length. 

MASSACHUSETTS  INSTITUTE  OF  TECHNOLOGY, 
Boston,  Mass.,  May,  f8g8. 


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MAY    7  1337 

AUG  10   1937    . 

WOV  24  I83g 

•* 

o    1948 

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D  0  D 

*AR  S     195P  '  n 

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LD  21-100? 

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me  •  : 
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